Senior executives at the News of the World, including Rebekah Brooks and Andy Coulson, did not lift a finger to stop hacking at the paper after Milly Dowler's phone messages were intercepted in 2002, the phone-hacking trial has heard.

The interception of Dowler's voicemails is "an extremely important" part of the prosecution case against Brooks, Coulson and the paper's then managing editor Stuart Kuttner that "fixes" them to "knowledge of phone hacking", jurors were told.

Brooks was editor at the time, but Coulson, her deputy, was at the helm when the paper despatched a squad of reporters to Telford to follow a lead that the 13-year-old was going for a job in a computer factory.

In his closing speech, prosecutor Andrew Edis QC said the Dowler hacking was no secret at the paper because bosses believed it was in the public interest.

He reminded jurors they had heard that Kuttner had told police about it and that the readers would also have known of the hack because the message was mentioned in the first edition of the story.

"The only explanation [for it not being a secret] is because the phone hacking was being used to try and find Milly Dowler, nobody would mind," said Edis.

"Even if that's wrong, from April 2002, the three defendants who are charged with count 1 on this indictment all knew about phone hacking and what is common ground is that none of them lifted a finger to stop it.

"Nobody got told off or sacked, the police were not called, despite the fact that at this stage everyone knew that a phone hack had been done by News of the World," said Edis.

He said the "tragic disappearance" of Dowler in March 2002 "was an enormous tragedy and it was therefore of great and proper interest to the newspaper. There was nothing wrong with them taking an interest in that story and they certainly did."

Edis told the jury that Coulson and Brooks would have discussed the Dowler story as it could have replaced an interview with EastEnders star Michael Greco on the front page.

"Both editors accepted in their evidence that if they had found her, that would have been the front-page story that week," he said.

"We say that is an extremely important part of the case, that fixes Mrs Brooks, Mr Coulson and Mr Kuttner with knowledge of phone hacking at the very latest April 2002."

Edis argued that Coulson's account of his knowledge of the hacking of David Blunkett's voicemails two years later was "falsified".

Coulson had said he was very "angry" when his reporter Neville Thurlbeck told him of the messages that suggested Blunkett was having an affair with a married woman. The reality, Edis claimed, was different. "It was 'carry on hacking at the News of the World', wasn't it?"

The prosecution also asked the jury to take a view on the veracity of Kuttner's claim that he couldn't remember events.

Edis said Kuttner said his memory was impaired by illness, including a brain stem stroke, and that he had it tested in 2013 but that expert analysis was not entered as evidence in the trial.

"So you only have his word that there's anything the matter with his brain at all," said Edis.

He also asked the jury to remember that when Kuttner was giving evidence, "he remembered when he thought it would help him", such as the case of David Shayler, the former MI5 whistleblower.

"Whether or not there is anything the matter with Mr Kuttner's memory, you are going to have to decide – because that is actually his case, he can't remember," said Edis.

Closing the prosecution's case, Edis told the jury they should listen to the defence speeches that are now to follow with the "same sceptical care to which you listened to me".

He told them: "We are here to set out our cases – none of us can tell you what to do … if you approach that in a dispassionate and fair way, then nobody in this court can complain. because that is your job."

"Nobody pretends that you have an an easy task," he said.

Kuttner, Brooks and Coulson have all been charged with conspiring to hack phones. They deny all charges against them.